Subgroups — C₂×#11727638 #13506454

Order 1792.

738 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#405	C₂	1	1	yes	maximal, characteristic
2	C₂²×#6386	C₂	1	2	yes	maximal
3	#11727638	C₂	1	8	yes	maximal, direct product factor
4	C₂×#89261	C₂	1	4	yes	maximal, complemented
5	C₂²×C₇×M₁₆	C₄	1	1	yes	characteristic
6	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₇×#117	C₄	1	1	yes	characteristic
8	C₇×#405	C₄	1	4	yes	—
9	C₇×#405	C₂²	1	4	yes	—
10	C₂×C₇×#97	C₂²	1	2	yes	—
11	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
12	C₂×#6386	C₄	1	4	yes	complemented
13	C₂×#6386	C₄	1	8	yes	complemented
14	C₂×#6386	C₂²	1	4	yes	—
15	C₂×#6386	C₂²	1	8	yes	—
16	C₂²×C₇⋊C₁₆	C₄	1	4	yes	complemented
17	#89261	C₂²	1	16	yes	complemented
18	C₂×#2110	—	7	1	no	maximal, Sylow
19	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₇×M₁₆	—	2	2	no	—
21	C₂×C₇×M₁₆	—	2	2	no	—
22	C₂×C₇×M₁₆	C₂×C₄	1	4	yes	—
23	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
29	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
30	C₂²×C₇×C₈	C₂³	1	2	yes	—
31	C₇×#117	—	2	2	no	—
32	C₇×#117	C₂×C₄	1	4	yes	—
33	C₂×C₇×C₄⋊C₄	—	2	1	no	—
34	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
35	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
36	C₇×#97	C₂×C₄	1	4	yes	—
37	C₇×#97	C₂³	1	4	yes	—
38	C₄×C₇×C₈	C₂×C₄	1	8	yes	—
39	#6386	C₂×C₄	1	8	yes	—
40	#6386	C₂×C₄	1	16	yes	complemented
41	#6386	C₂³	1	8	yes	—
42	C₂×C₇⋊C₁₆	C₂×C₄	1	8	yes	—
43	C₂×C₇⋊C₁₆	C₂×C₄	1	16	yes	complemented
44	C₂²×M₃₂	—	7	2	no	—
45	C₂×#405	D₇	1	1	yes	characteristic
46	#2110	—	7	8	no	—
47	C₂×#347	—	7	4	no	—
48	C₂⁴×C₇	M₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₇	—	2	2	no	—
50	C₂²×C₄×C₇	C₄²	1	2	yes	—
51	C₂²×C₄×C₇	C₄²	1	2	yes	—
52	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₇	M₁₆	1	2	yes	—
54	C₂²×C₄×C₇	M₁₆	1	2	yes	—
55	C₂²×C₄×C₇	M₁₆	1	2	yes	—
56	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
58	C₇×M₁₆	—	2	8	no	—
59	C₂×C₇×C₈	—	2	2	no	—
60	C₂×C₇×C₈	—	2	2	no	—
61	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
62	C₂×C₇×C₈	C₄²	1	2	yes	—
63	C₂×C₇×C₈	C₄²	1	4	yes	—
64	C₂×C₇×C₈	C₄²	1	4	yes	—
65	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
66	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
67	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
68	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
70	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
71	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
72	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
73	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
74	C₇×C₄⋊C₄	—	2	4	no	—
75	C₇×C₂²⋊C₄	—	2	4	no	—
76	C₄²×C₇	—	2	2	no	—
77	C₄²×C₇	C₂²×C₄	1	4	yes	—
78	C₇⋊C₁₆	C₂²×C₄	1	16	yes	—
79	C₂²×M₁₆	Dic₇	1	1	yes	characteristic
80	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
81	C₂×#117	Dic₇	1	1	yes	characteristic
82	C₂×M₃₂	—	7	4	no	—
83	C₂×M₃₂	—	7	8	no	—
84	C₂×M₃₂	—	7	4	no	—
85	C₂×M₃₂	—	7	8	no	—
86	C₂²×C₁₆	—	7	4	no	—
87	#405	Dic₇	1	4	yes	—
88	#405	C₂×D₇	1	4	yes	—
89	C₂×#97	C₂×D₇	1	2	yes	—
90	C₂×C₄×C₈	Dic₇	1	2	yes	—
91	#347	—	7	16	no	—
92	C₂³×C₇	—	4	2	no	—
93	C₂³×C₇	#97	1	2	yes	—
94	C₂³×C₇	#97	1	2	yes	—
95	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
96	C₂³×C₇	C₂×M₁₆	1	2	yes	—
97	C₂×C₄×C₇	—	2	4	no	—
98	C₂×C₄×C₇	—	4	4	no	—
99	C₂×C₄×C₇	#97	1	2	yes	—
100	C₂×C₄×C₇	#97	1	4	yes	—
101	C₂×C₄×C₇	#97	1	4	yes	—
102	C₂×C₄×C₇	#97	1	2	yes	—
103	C₂×C₄×C₇	C₂×C₄²	1	4	yes	—
104	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
109	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
111	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
112	C₇×C₈	—	2	4	no	—
113	C₇×C₈	C₂×C₄²	1	2	yes	—
114	C₇×C₈	C₂×C₄²	1	2	yes	—
115	C₇×C₈	C₂×C₄²	1	2	yes	—
116	C₇×C₈	C₂³×C₄	1	2	yes	—
117	C₂³×C₄	C₂×Dic₇	1	1	yes	characteristic
118	C₂×M₁₆	—	2	2	no	—
119	C₂×M₁₆	—	2	2	no	—
120	C₂×M₁₆	C₂×Dic₇	1	4	yes	—
121	C₂²×C₈	C₄×D₇	1	2	yes	—
122	C₂²×C₈	C₄×D₇	1	4	yes	—
123	C₂²×C₈	C₄×D₇	1	2	yes	—
124	C₂²×C₈	C₂×Dic₇	1	2	yes	—
125	C₂²×C₈	C₂×Dic₇	1	2	yes	—
126	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
128	C₂²×C₈	C₂²×D₇	1	2	yes	—
129	#117	—	2	2	no	—
130	#117	C₂×Dic₇	1	4	yes	—
131	C₂×C₄⋊C₄	—	2	1	no	—
132	C₂×C₂²⋊C₄	—	2	1	no	—
133	C₂×C₄²	C₂×Dic₇	1	2	yes	—
134	M₃₂	—	7	8	no	—
135	M₃₂	—	7	16	no	—
136	M₃₂	—	7	8	no	—
137	C₂×C₁₆	—	7	8	no	—
138	C₂×C₁₆	—	7	16	no	—
139	#97	C₂×Dic₇	1	4	yes	—
140	#97	C₂²×D₇	1	4	yes	—
141	C₄×C₈	C₂×Dic₇	1	8	yes	—
142	C₂²×C₇	—	2	2	no	—
143	C₂²×C₇	—	2	4	no	—
144	C₂²×C₇	—	4	2	no	—
145	C₂²×C₇	—	4	2	no	—
146	C₂²×C₇	C₂×#97	1	1	yes	characteristic
147	C₂²×C₇	C₂×#97	1	4	yes	—
148	C₂²×C₇	C₂×#97	1	1	yes	characteristic
149	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
150	C₄×C₇	—	4	4	no	—
151	C₄×C₇	C₂×#97	1	1	yes	characteristic
152	C₄×C₇	C₂×#97	1	1	yes	characteristic
153	C₄×C₇	C₂×#97	1	4	yes	—
154	C₄×C₇	C₂²×C₄²	1	1	yes	characteristic
155	C₄×C₇	C₂²×M₁₆	1	1	yes	characteristic
156	C₂⁴	#1136	1	1	yes	characteristic
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	#1136	1	2	yes	—
159	C₂²×C₄	#1136	1	2	yes	—
160	C₂²×C₄	#1136	1	1	yes	characteristic
161	C₂²×C₄	#1136	1	2	yes	—
162	C₂²×C₄	C₄×Dic₇	1	2	yes	—
163	C₂²×C₄	C₄×Dic₇	1	2	yes	—
164	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
165	C₂²×C₄	C₂²×Dic₇	1	2	yes	—
166	M₁₆	—	2	8	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	C₄×Dic₇	1	4	yes	—
170	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
171	C₂×C₈	C₄×Dic₇	1	2	yes	—
172	C₂×C₈	C₄×Dic₇	1	4	yes	—
173	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
174	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
175	C₂×C₈	C₂×C₄×D₇	1	4	yes	—
176	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
177	C₂×C₈	C₂×C₄×D₇	1	4	yes	—
178	C₂×C₈	C₂²×Dic₇	1	4	yes	—
179	C₂×C₈	C₂²×Dic₇	1	2	yes	—
180	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
181	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
182	C₄⋊C₄	—	2	4	no	—
183	C₂²⋊C₄	—	2	4	no	—
184	C₄²	—	2	2	no	—
185	C₄²	C₂²×Dic₇	1	4	yes	—
186	C₁₆	—	7	16	no	—
187	C₂×C₇	—	2	1	no	—
188	C₂×C₇	—	2	1	no	—
189	C₂×C₇	—	2	4	no	—
190	C₂×C₇	#2110	1	2	yes	complemented
191	C₂×C₇	C₂²×#97	1	1	yes	characteristic
192	C₂³	—	4	2	no	—
193	C₂³	#6377	1	2	yes	—
194	C₂³	#6377	1	2	yes	—
195	C₂³	C₂×#1136	1	1	yes	characteristic
196	C₂³	C₂×#1136	1	2	yes	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	#6377	1	2	yes	—
200	C₂×C₄	#6377	1	4	yes	—
201	C₂×C₄	#6377	1	4	yes	—
202	C₂×C₄	#6377	1	2	yes	—
203	C₂×C₄	C₂×#1136	1	1	yes	characteristic
204	C₂×C₄	C₂×#1136	1	2	yes	—
205	C₂×C₄	C₂×#1136	1	1	yes	characteristic
206	C₂×C₄	C₂×#1136	1	1	yes	characteristic
207	C₂×C₄	C₂×#1136	1	4	yes	—
208	C₂×C₄	C₂×C₄×Dic₇	1	4	yes	—
209	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
211	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
212	C₈	—	2	4	no	—
213	C₈	C₂×C₄×Dic₇	1	2	yes	—
214	C₈	C₂×C₄×Dic₇	1	2	yes	—
215	C₈	C₂×C₄×Dic₇	1	2	yes	—
216	C₈	C₂²×C₄×D₇	1	2	yes	—
217	C₇	C₂×#2110	1	1	yes	characteristic, complemented, Sylow
218	C₂²	—	2	2	no	—
219	C₂²	—	2	4	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	C₂×#6377	1	1	yes	characteristic
223	C₂²	C₂×#6377	1	4	yes	—
224	C₂²	C₂×#6377	1	1	yes	characteristic
225	C₂²	C₂²×#1136	1	1	yes	characteristic
226	C₄	—	4	4	no	—
227	C₄	C₂×#6377	1	1	yes	characteristic
228	C₄	C₂×#6377	1	1	yes	characteristic
229	C₄	C₂×#6377	1	4	yes	—
230	C₄	C₂²×#1136	1	1	yes	characteristic
231	C₄	C₂²×C₄×Dic₇	1	1	yes	characteristic
232	C₂	—	2	1	no	—
233	C₂	—	2	1	no	—
234	C₂	—	2	4	no	—
235	C₂	#11727638	1	2	yes	direct product factor
236	C₂	C₂²×#6377	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×Dic₇
commutator	C₄×C₇	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂×C₇×#405	C₂
socle	C₂²×C₇	C₂×#97
prime radical	C₂⁴×C₇	M₁₆
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	C₂×#405	D₇
hypocenter	C₇	C₂×#2110
LCS	G ▷ C₄×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #11727638	C₂	direct (N × H)
4. C₂×#89261	C₂	semidirect
12. C₂×#6386	C₄	semidirect
13. C₂×#6386	C₄	semidirect
16. C₂²×C₇⋊C₁₆	C₄	semidirect
17. #89261	C₂²	semidirect
40. #6386	C₂×C₄	semidirect
43. C₂×C₇⋊C₁₆	C₂×C₄	semidirect
190. C₂×C₇	#2110	semidirect
217. C₇	C₂×#2110	semidirect
235. C₂	#11727638	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₇×#405	2
C₂²×#6386	2
#11727638	2
C₂×#89261	2
C₂×#2110	7

Subgroups — C2×#11727638 #13506454

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11727638 #13506454